During application of a surgical knife or scalpel bleeding can be reduced by cauterizing the cut tissue through heat. According to the prior art, this can be obtained, for example, by applying resistance elements near the cutting edge of the scalpel which is electrically heated to provide a temperature of 200.degree.-500.degree. C. in contact with the tissue. In doing so, however, parts of the blade not in contact with tissue may become grossly overheated, presenting a hazard to both patient and surgeon. Several methods have been recommended to overcome this problem. Generally, all have disadvantages. For example, segmented heating elements require individual temperature-power control systems; heating elements with a large negative temperature coefficient of resistance (TCR) require a very high voltage drive; and, high frequency electric discharge applied through the patient's body produces bad scars and is hard to control.